L(s) = 1 | + (−1.56 + 2.70i)5-s + (−17.1 − 6.88i)7-s + (−33.2 + 19.2i)11-s − 13.8i·13-s + (−47.5 − 82.4i)17-s + (−9.86 − 5.69i)19-s + (−23.9 − 13.8i)23-s + (57.6 + 99.7i)25-s + 44.2i·29-s + (119. − 68.7i)31-s + (45.4 − 35.7i)35-s + (70.6 − 122. i)37-s + 337.·41-s − 417.·43-s + (−145. + 251. i)47-s + ⋯ |
L(s) = 1 | + (−0.139 + 0.242i)5-s + (−0.928 − 0.371i)7-s + (−0.912 + 0.526i)11-s − 0.294i·13-s + (−0.678 − 1.17i)17-s + (−0.119 − 0.0687i)19-s + (−0.217 − 0.125i)23-s + (0.460 + 0.798i)25-s + 0.283i·29-s + (0.690 − 0.398i)31-s + (0.219 − 0.172i)35-s + (0.313 − 0.543i)37-s + 1.28·41-s − 1.48·43-s + (−0.450 + 0.779i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.173877694\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.173877694\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (17.1 + 6.88i)T \) |
good | 5 | \( 1 + (1.56 - 2.70i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (33.2 - 19.2i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 13.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (47.5 + 82.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (9.86 + 5.69i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (23.9 + 13.8i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 44.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-119. + 68.7i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-70.6 + 122. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 417.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (145. - 251. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (14.7 - 8.53i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-299. - 519. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-459. - 265. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (325. + 563. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 934. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-787. + 454. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-397. + 688. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 314.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-179. + 310. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 80.5iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.698922484169278503649666845333, −8.951075514514457233814748018920, −7.78745040578032824950754558890, −7.15826622414169573889140365899, −6.36958659640791987600627139543, −5.27653764914077167749770418550, −4.36231766556239065601043612547, −3.18761515797906879956766867311, −2.39856413639227193024886382414, −0.64562504323129142172065599789,
0.47290021270009675045153971296, 2.11115976604898263688676962868, 3.13700667574877068606210121906, 4.16356005796456409839664085567, 5.23640235095433872056136347885, 6.20438322903104948193993047898, 6.78919546544416039759350881436, 8.178429592904393154283209655687, 8.491726357995147369255798686231, 9.594472176556475106294325447870